Answer:
(x+2), (x-4), (x-3)
Step-by-step explanation:
Use the rational root theorem to get started, then factor the remaining quadratic to find:
x^3 − 5x^2 − 2x + 24 = (x + 2)(x − 4)(x − 3)
Explanation:
Let f(x) =x^3 − 5x^2 − 2x + 24
By the rational root theorem, any rational zeros of f(x) must be expressible in the for p/q for integers p, q with p a divisor of the constant term 24 and q a divisor of the coefficient 1 of the leading term.
That means that the only possible rational zeros are the factors of 24, namely:
± 1, ± 2, ± 3, ± 4, ± 6, ± 12, ± 24
Try each in turn:
f(1) = 1 − 5 − 2 + 24 = 18
f(−1) = −1 − 5 + 2 + 24 = 20
f(2) = 8 − 20 − 4 + 24 = 8
f(−2) = −8 − 20 + 4 + 24 = 0
So x = −2 is a zero and (x + 2) is a factor.
x^3 − 5x^2 − 2x + 24 = (x + 2)(x^2 − 7x + 12)
We can factor
x^2 − 7x + 12 by noting that 4 × 3 = 12 and 4 + 3 = 7, so:
x^2 − 7x + 12 = (x − 4)(x − 3)
Putting it all together:
x^3 − 5x^2 − 2x + 24 = (x + 2)(x − 4)(x − 3)
